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Chapter 7: Q6.2-18E (page 395)

Question: In Exercises 17-22, determine which sets of vectors are orthonormal. If a set is only orthogonal, normalize the vectors to produce an orthonormal set.

18. \(\left( {\begin{array}{*{20}{c}}0\\1\\0\end{array}} \right),{\rm{ }}\left( {\begin{array}{*{20}{c}}0\\{ - 1}\\0\end{array}} \right)\)

Short Answer

Expert verified

The set of vectors \(\left\{ {{\bf{u}},{\bf{v}}} \right\}\) is not orthogonal because \({\bf{u}} \cdot {\bf{v}} \ne 0\).

Step by step solution

01

Definition of orthonormal sets

A set \(\left\{ {{{\mathop{\rm u}\nolimits} _1},{{\mathop{\rm u}\nolimits} _2}, \ldots ,{{\mathop{\rm u}\nolimits} _p}} \right\}\) is called anorthonormal setwhen it is an orthogonal set of unit vectors.

02

Check whether the set of vectors are orthogonal

Consider that \({\bf{u}} = \left( {\begin{array}{*{20}{c}}0\\1\\0\end{array}} \right),{\rm{ }}{\bf{v}} = \left( {\begin{array}{*{20}{c}}0\\{ - 1}\\0\end{array}} \right)\).

Check whether the set of the vector is orthogonal, as shown below:

\(\begin{array}{c}{\bf{u}} \cdot {\bf{v}} = \left( {\begin{array}{*{20}{c}}0\\1\\0\end{array}} \right) \cdot \left( {\begin{array}{*{20}{c}}0\\{ - 1}\\0\end{array}} \right)\\ = 0\left( 0 \right) + 1\left( { - 1} \right) + 0\left( 0 \right)\\ = 0 - 1 + 0\\ = - 1\\ \ne 0\end{array}\)

It is observed that \(\left\{ {{\bf{u}},{\bf{v}}} \right\}\) is not an orthogonal set because \({\bf{u}} \cdot {\bf{v}} \ne 0\).

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Most popular questions from this chapter

Let \(A = PD{P^{ - {\bf{1}}}}\), where P is orthogonal and D is diagonal, and let \(\lambda \) be an eigenvalue of A of multiplicity k. Then \(\lambda \) appears k times on the diagonal of D.Explain why the dimension of the eigenspace for \(\lambda \) is k.

(M) Compute an SVD of each matrix in Exercises 26 and 27. Report the final matrix entries accurate to two decimal places. Use the method of Examples 3 and 4.

27. \(A{\bf{ = }}\left( {\begin{array}{*{20}{c}}{\bf{6}}&{ - {\bf{8}}}&{ - {\bf{4}}}&{\bf{5}}&{ - {\bf{4}}}\\{\bf{2}}&{\bf{7}}&{ - {\bf{5}}}&{ - {\bf{6}}}&{\bf{4}}\\{\bf{0}}&{ - {\bf{1}}}&{ - {\bf{8}}}&{\bf{2}}&{\bf{2}}\\{ - {\bf{1}}}&{ - {\bf{2}}}&{\bf{4}}&{\bf{4}}&{ - {\bf{8}}}\end{array}} \right)\)

In Exercises 17–24, \(A\) is an \(m \times n\) matrix with a singular value decomposition \(A = U\Sigma {V^T}\) , where \(U\) is an \(m \times m\) orthogonal matrix, \({\bf{\Sigma }}\) is an \(m \times n\) “diagonal” matrix with \(r\) positive entries and no negative entries, and \(V\) is an \(n \times n\) orthogonal matrix. Justify each answer.

21. Justify the statement in Example 2 that the second singular value of a matrix \(A\) is the maximum of \(\left\| {A{\bf{x}}} \right\|\) as \({\bf{x}}\) varies over all unit vectors orthogonal to \({{\bf{v}}_{\bf{1}}}\), with \({{\bf{v}}_{\bf{1}}}\) a right singular vector corresponding to the first singular value of \(A\). (Hint: Use Theorem 7 in Section 7.3.)

Orthogonally diagonalize the matrices in Exercises 13–22, giving an orthogonal matrix \(P\) and a diagonal matrix \(D\). To save you time, the eigenvalues in Exercises 17–22 are: (17) \( - {\bf{4}}\), 4, 7; (18) \( - {\bf{3}}\), \( - {\bf{6}}\), 9; (19) \( - {\bf{2}}\), 7; (20) \( - {\bf{3}}\), 15; (21) 1, 5, 9; (22) 3, 5.

21. \(\left( {\begin{aligned}{{}}4&3&1&1\\3&4&1&1\\1&1&4&3\\1&1&3&4\end{aligned}} \right)\)

Question: Compute the singular values of the \({\bf{4 \times 4}}\) matrix in Exercise 9 in Section 2.3, and compute the condition number \(\frac{{{\sigma _1}}}{{{\sigma _4}}}\).
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